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sequence by a number (decimal), such as -000 .. . *000 *.,,in
which a correspondingly large number of zeros occur before
another digit (1? 2, ... or 9) appears.
This illustrates what is meant by a convergent sequence of
numbers: the rationals 1,1-4,*., constituting the sequence give
us ever closer approximations to the "irrational number' •which
we call the square root of 2, and which we conceive of as having
been defined by the convergent sequence of rationals, this defini-
tion being in the sense that a method has been indicated (the
usual school one) of calculating any particular member of the
sequence in a finite number of steps.
Although it is impossible actually to exhibit the whole
sequence, as it does not stop at any finite number of terms,
nevertheless we regard the process for constructing any member
of the sequence as a sufficiently clear conception of the whole
sequence as a single definite object which we can reason about.
Doing so, we have a workable method for using the square root
of 2 and similarly for any irrational number, in mathematical
As has been indicated it is impossible to make this precise in
an account like the present, but even a careful statement might
disclose some of the logical objections glaringly apparent in the
above description - objections which inspired Kronecker and
others to attack Weierstrass' 'sequential' definition of
Nevertheless, right or wrong, Weierstrass and his school
made the theory work. The most useful results they obtained
have not yet been questioned, at least on the ground of their
great utility in mathematical analysis and its applications, by
any competent judge in his right mind. This does not mean that
objections cannot be well taken: it merely calls attention to the
fact that in mathematics, as in everything else, this earth is not
yet to be confused with the Kingdom of Heaven, that perfection
is a chimaera, and that, in the words of Crelle, we can only hope
for closer and closer approximations to mathematical truth -
whatever that may be, if anything - precisely as in the Weier-
strassian theory of convergent sequences of rationals defhiing